Short answer: no.
Long answer: a qualified no...
Analog fax machines can't interface with digital cellphones, mostly
due to the timing delays caused by the audio compression. If your
carrier still allows analog phones to be activated, you could search
eBay and thift stores for old analog phones that have a data/fax
interface (the interface has a regular RJ11 phone jack to plug a fax
into.) I activated an old Nokia phone on pre-paid with Cingular last
summer for use with a cdit-card terminal and a fax machine.
Connecting the cellphone to a computer to allow fax software to
receive faxes is easier.
> I have a landline but whenever I receive a fax message my
> parents keep lifting the phone to answer.
How about just training your parents? Most fax machines have a manual
"forced answer" mode. If I pick up my phone to find it's an incoming
fax, I can hit #1 on the phone keypad to force my fax to "wake up"
then I can hang up and let the fax machine receive.
Optionally, you could add "distinctive ring" to your landline. For a
few bucks a month you get a second phone number (not a second phone
line) than rings in a different "pattern" (long-short, short-short,
etc.). Use the new number as your fax number and tell your parents
not to answer the "funny" rings.